1. Field of the Invention
The present invention relates to a method for modelling two-phase or three-phase flows in a porous medium, in drainage and imbibition. It is based on a fractal representation of the porous medium and on an original approach for handling phenomena linked with hysteresis (change in the direction of variation of the saturations).
2. Description of the Prior Art
1) Experimental Studies
Experimental determination of the relative permeabilities of a porous medium wherein a multiphase fluid flows is not easy. Measuring operations are usually simplified by considering that one of the phases is immobile in a state of irreducible saturation.
The values are for example acquired by means of a well-known experimental method referred to as “steady state” for determining relative permeabilities, which allows a three-phase fluid to flow with imposed flow rates between the phases. The relative permeabilities expressed as a function of the two saturations are calculated by applying Darcy's law to each phase. It is not an established fact that the relative permeability measurements obtained by means of this method are really representative of the fluid displacements and, in any case, they take a long time because, at each regime change, one has to wait for a state of equilibrium.
Another known method carries out laboratory tests in order to determine measurement tables (as shown in FIG. 1) relating the relative permeabilities and the saturations for each pair of fluids of the three-phase mixture. By adjusting experimental production curves, one tries to progressively adjust the three-phase relative permeabilities. These data tables are then entered into an Athos® type numerical simulator which computes the fluid productions. This method is based on the prior acquisition of many experimental measurements progressively adjusted by calibration and takes a long time.
2) Relative Permeability Models
The known empirical model referred to as Stone's model allows, by empirical correlations, to predict data relative to a three-phase flow from data corresponding to a two-phase flow. It is valid only in case of a high water wettability and it is generally considered to be a poor predictor.
There are two known types of physical models for modelling three-phase flows, based on capillary pressure curves. The capillary pressure curves are connected with a saturation (for example that of the mercury injected) and a pore radius from which the mercury stops for a given injection pressure, determined by Laplace's law, Pinj.
      Pinj    .    =                    2        ⁢        σ            r        .  
A first porous media representation model is described by:
Burdine, N. T.: “Relative Permeability Calculations from Pore Size Distribution Data”, Trans AIME (1953), Vol. 198, or by
Corey, A. T.: “The Interrelation between Oil and Gas Relative Permeabilities”, Prod. Monthly (1954), Vol. 19, 38.
According to this model, the porous medium is represented by a bundle of cylindrical capillaries with a radius distribution given by the capillary pressure curve obtained by mercury injection. The permeabilities are obtained by applying Poiseuille's law to the flow of fluids in these capillaries.
This model is based on the representation of the porous medium as an assembly of capillaries with different radii. The relation between the volume and the radius of the pores is given by the value of the slope of the pseudo-plateau. The three fluids are supposed to share the capillaries between them, the wetting fluid (water) occupying the smallest ones, the least wetting fluid (gas) the largest ones and the third fluid (oil) a zone with intermediate-size pores. It is not possible to describe the interactions between the fluids because, in such a model, they flow through separate channels. Finally, this model can be useful only if the pseudo-plateau covers a wide range of saturations. According to this model, the three phases of a three-phase flow move in different capillaries and there is no interaction between them.
Another known physical porous medium representation model is described by:
de Gennes, P. G.: “Partial Filling of a Fractal Structure by a Wetting Fluid”, Physics of Disordered Materials 227–241, New York Plenum Pub. Corp. (1985), taken up by
Lenormand, R.: “Gravity Assisted Inert Gas Injection: Micromodel Experiments and Model based on Fractal Roughness”, The European Oil and Gas Conference Altavilla Milica, Palermo, Sicily (1990).
According to this model, the inner surface of the pores is considered isotropic and has a fractal character, and it can be modelled as a “bunch” of parallel capillary grooves so that the pores exhibit a fractal cross section. The cross section of each pore is constructed according to an iterative process (FIG. 1). The half-perimeter of a circle of radius R0 is divided into η parts and each of these η parts is replaced by a semi-circle or groove. At each stage k of the process, Nk new semi-circular grooves of radius Rk and of total section Ak are created.
The fractal dimension DL of the cross section at the end of stage k is related to the number of objects Nk generated with the given scale Ik by the relation:Nk∞Ik−DL 
The fractal dimension can be deduced from a mercury capillary pressure curve according to the following procedure. Mercury is injected into a porous medium with an injection pressure that increases in stages. Laplace's law allows deducing the pore volume, knowing the volume of mercury injected for a given capillary pressure and the drainage capillary pressure curve relating the injection pressure to the amount of mercury injected and the curve relating the proportion of the total volume occupied by the pores and the size of the pores can be constructed. In cases where a wetting liquid is drained from the porous medium such as water by gas injection, the correlation between the gas-water capillary pressure and the saturation of the wetting phase is given by:
      P    C    =      S    W          1                        D          L                -        2            
The experimental results readily show that the values of the gas-water relative permeabilities expressed as a function of the three saturations, obtained from the expressions given by the known models and the phase distribution modes in the structure of the pores, are far from the measured values and therefore that the models concerned prove to be too simplistic to represent the complex interactions that take place between the fluid phases.
French Patent 2,772,483 (U.S. Pat. No. 6,021,662) describes a modelling method for optimizing faster and more realistically the flow conditions, in a porous medium wettable by a first fluid (water for example), of a mixture of fluids including this wetting fluid and at least another fluid (oil and possibly gas). This method involves modelling the pores of the porous medium by a distribution of capillaries with a fractal distribution considering, in the case of a three-phase water (wetting fluid)-oil-gas mixture for example, a stratification of the constituents in the pores, with the water in contact with the walls, the gas in the center and the oil forming an intercalary layer. It comprises experimental determination of the variation curve of the capillary pressure in the pores as a function of the saturation in the liquid phases, from which the fractal dimension values corresponding to a series of given values of the saturation in the liquid phase are deduced. It also comprises modelling the relative permeabilities directly in a form of analytic expressions depending on the various fractal dimension values obtained and in accordance with the stratified distribution of the different fluids in the pores. A porous medium simulator is used from these relative permeabilities to determine the optimum conditions of displacement of the fluids in the porous medium.
The hysteresis phenomenon relates to the variations in the petrophysical properties (relative permeabilities, capillary pressure, resistivity index, etc.) observed according to whether measurements are performed during drainage or imbibition (these modes respectively correspond to a saturation increase and decrease of the non-wetting phase). This phenomenon must therefore be taken into account to obtain representative relative permeability values.
The prior art concerning hysteresis effects in two-phase and three-phase media is described for example in the following publications:
Land C. S.: “Calculation of Imbibition Relative Permeability for Two and Three-Phase Flow from Rock Properties”, Trans AIME 1968, Vol. 243, 149,
Larsen J. A., Skauge A.: “Methodology for Numerical Simulation with Cycle-Dependent Relative Permeability”, SPEJ, June 1998, and
Carlson F. M.: “Simulation of Relative Permeability Hysteresis to the Non-Wetting Phase”, SPE 10157, ATCE, San Antonio, Tex., 4–7 Oct. 1981.
FIG. 6 typically shows the course of the two-phase permeability curves Krw resulting from drainage up to irreducible saturation in wetting fluid (M), then imbibition up to residual saturation in non-wetting fluid (NM). The hysteresis phenomenon occurs at two levels. At equal saturations Sg, different numerical values are obtained and the end point reached is an unknown that depends on the cusp point SgM from which the displacement mode is changed. This phenomenon is usually attributed to the trapped non-wetting fluid fraction. At equal saturations, the same quantity of mobile fluid is not obtained, which distorts the flow characteristics.
Practically all the models taking account of hysteresis effects involve Land's semi-empirical relation:
                                          1                          S              gr                                -                      1                          S              gi                                      =                  C          L                                    (        1        )            where CL represents Land's constant. This relation relates the initial saturation Sgi to the residual saturation Sgr in non-wetting fluid, in order to evaluate the saturations in trapped and free non-wetting fluid. Assuming that this relation is valid whatever the saturation, it is applied to determine the intermediate mobile fractions during displacement. In the case of two-phase flows, associating this relation with permeability models provides satisfactory results.
In the case of three-phase flows, the hysteresis of the relative permeabilities Krg takes on a particular form. A displacement hysteresis is experimentally observed, as it is the case with two-phase flows (extent of the direction of variation of the saturations), as well as a cycle hysteresis since the permeabilities depend on the saturation record. In FIG. 7, the relative permeability curves Krg corresponding to a first drainage and imbibition cycle (D1 and I1 respectively) are distinct from the corresponding curves (D2, I2) of a second cycle.
The relative permeability model developped by Larsen et al takes these two forms of hysteresis into account. Starting from an approach combining Stone's model in parallel with Land's formula and Carlson's interpolation method, an approach where only the displacement hysteresis is taken into account, Larsen et al have introduced an empirical reduction factor that is a function of the water saturation, which allows to approximate to the permeability reduction of the gas associated with the cycle hysteresis.